Finite Group Databasebeta

← #24908814

Subgroups — #24908814

Order 10624.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128332C211yesmaximal, characteristic
2#19127743C212yesmaximal
3#19127566C211yesmaximal, characteristic
4#19127462C212yesmaximal
5#343×C83C211yesmaximal, characteristic
6#14581791—22no—
7#14581783—22no—
8#14581771—22no—
9#14581751C2212yes—
10C2×C4⋊Dic83C2211yescharacteristic
11C2×C4×Dic83C2211yescharacteristic
12#14581726—22no—
13#14581724—22no—
14C83⋊#95—22no—
15C2×M16×C83C2212yes—
16C2×C4⋊C4×C83C2211yescharacteristic
17C2×Dic166—22no—
18C22×Dic83—21no—
19C22×Dic83—21no—
20C4⋊Dic83—22no—
21C4⋊Dic83—22no—
22C4⋊Dic83D412yes—
23C4⋊Dic83D412yes—
24Dic83⋊C4—24no—
25C22⋊Dic83—22no—
26C4×Dic83—22no—
27C4×Dic83—21no—
28C4×Dic83Q812yescomplemented
29C22×C4×C83D412yes—
30C22×C4×C83C2311yescharacteristic
31M16×C83—22no—
32M16×C83Q814yescomplemented
33C2×C8×C83—22no—
34C4⋊C4×C83—22no—
35C23×C83C2×D411yescharacteristic
36C2×C4×C83—22no—
37C2×C4×C83—42no—
38C2×C4×C83C2×D412yes—
39C2×C4×C83C2×Q811yescharacteristic
40C2×C4×C83C2×Q812yes—
41C2×C4×C83#4111yescharacteristic
42C2×Dic83—22no—
43C2×Dic83—22no—
44C2×Dic83—22no—
45C2×Dic83—41no—
46C2×Dic83—41no—
47C2×Dic83—42no—
48C2×Dic83—41no—
49Dic166—82no—
50C8×C83—24no—
51C22×C83—21no—
52C22×C83—21no—
53C22×C83#12011yescharacteristic
54C22×C83C22⋊Q811yescharacteristic
55C22×C83C4⋊Q811yescharacteristic
56C4×C83—42no—
57C4×C83C22⋊Q812yes—
58C4×C83C4⋊Q812yes—
59Dic83—42no—
60Dic83—42no—
61Dic83—42no—
62Dic83—81no—
63C2×C83—21no—
64C2×C83—21no—
65C2×C83#39511yescharacteristic
66C2×C83#45411yescharacteristic
67C2×C83#45511yescharacteristic
68#2061—831nomaximal, Sylow
69C83#206111yescharacteristic, complemented, Sylow
70#544—831no—
71#428—832no—
72#382—831no—
73#343D8311yescharacteristic
74#339—832no—
75C2×M16C2×D8312yes—
76C4⋊Q8—1662no—
77#125—1662no—
78C22⋊Q8—1662no—
79#117—832no—
80C2×C4⋊C4—831no—
81C2×C4⋊C4C2×D8311yescharacteristic
82C2×C42—831no—
83#107—1662no—
84#106—1662no—
85#95—1662no—
86C2×Q8—1662no—
87C22×C4—1661no—
88C22×C4—1661no—
89C22×C4D33212yes—
90C22×C4C22×D8311yescharacteristic
91M16—22no—
92M16Dic16614yescomplemented
93C2×C8—22no—
94C4⋊C4—22no—
95C4⋊C4—832no—
96C4⋊C4—832no—
97C4⋊C4—1662no—
98C4⋊C4—1664no—
99C4⋊C4—1662no—
100C22⋊C4—1662no—
101C42—832no—
102C42—1662no—
103C42—1661no—
104C23C2×D33211yescharacteristic
105Q8—6642no—
106C2×C4—22no—
107C2×C4—42no—
108C2×C4—3321no—
109C2×C4—3321no—
110C2×C4—3322no—
111C2×C4—3321no—
112C2×C4—1662no—
113C2×C4—1662no—
114C2×C4—1662no—
115C2×C4D4×D8312yes—
116C2×C4Q8×D8311yescharacteristic
117C2×C4C2×Dic16612yes—
118C2×C4#1305058911yescharacteristic
119C8—24no—
120C22—21no—
121C22—21no—
122C22#1458176211yescharacteristic
123C22#1458177211yescharacteristic
124C22#1458179211yescharacteristic
125C4—42no—
126C4—3322no—
127C4—3322no—
128C4—3322no—
129C4—6641no—
130C4#1458176912yes—
131C4#1458179112yes—
132C2—21no—
133C2—21no—
134C2#1912762211yescharacteristic
135C2#1912787611yescharacteristic
136C2#1912788111yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#343×C83C2
socleC22×C83#120
prime radicalC23×C83C2×D4
hypercenter#343D83
hypocenterC83#2061
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
28. C4×Dic83Q8semidirect
32. M16×C83Q8semidirect
69. C83#2061semidirect
92. M16Dic166semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#191283322
#191277432
#191275662
#191274622
#343×C832
#206183